Is a learned policy, for a deterministic problem, trained in a supervised process, a stochastic policy?
Is the policy (based in the neural network) a stochastic policy? even if the action space is discrete?

Yes. A discrete action space does not require a deterministic policy - it is possible to assign arbitrary probabilities to each action in each state provided each probability is in the range $[0,1]$ and the sum across all allowed actions is $1$. The two concepts of determinism and discrete actions are entirely separate.
The optimal policy in many situations can be deterministic. If there is only one deterministic optimal policy, your learned policy should be also close to deterministic if the learning process has been successful. That is, the probabilities of optimal actions should all be close to $1$, all the rest close to $0$.
If there is more than one possible optimal policy, your learning agent may have learned a stochastic &quot;mix&quot; of them where in some states it is equally good to take more than one action and the probabilities may be split between those good actions. This will still be optimal and not a problem. If that is the case you should expect to see many action choices close to $0$ and in each state a select few (maybe one) that sum to close to $1$ between them.
In the case of discrete actions, you can derive a deterministic policy from your neural network function by taking the argmax of the action probabilities. This is worth trying. It will round away bad actions that are close to 0 probability due to approximation in the neural network.
In practice sometimes a little randomness in a policy works better for real-world problems with imprecise measurements or other unknowns. It may even be necessary for adversarial environments or where there is key information missing. The only way to find out though is to try both stochastic and deterministic interpretations of the neural network output for your policy.